Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, q\neq 0$. $\dfrac{{(p^{-4}q^{-1})^{3}}}{{(p^{2}q^{5})^{-2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{-4}q^{-1})^{3} = (p^{-4})^{3}(q^{-1})^{3}}$ On the left, we have ${p^{-4}}$ to the exponent ${3}$ . Now ${-4 \times 3 = -12}$ , so ${(p^{-4})^{3} = p^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{-4}q^{-1})^{3}}}{{(p^{2}q^{5})^{-2}}} = \dfrac{{p^{-12}q^{-3}}}{{p^{-4}q^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-12}q^{-3}}}{{p^{-4}q^{-10}}} = \dfrac{{p^{-12}}}{{p^{-4}}} \cdot \dfrac{{q^{-3}}}{{q^{-10}}} = p^{{-12} - {(-4)}} \cdot q^{{-3} - {(-10)}} = p^{-8}q^{7}$